The present invention relates generally to communication apparatus, and more particularly, to methods that provide for constellation pruning and partial Gray coding of the pruned constellation that may be used with a transmitter hardware architecture that implements two-dimensional, uncoded signaling for equiprobable signals suitable for multiple gigabit/second satellite communications.
At gigabit/second and higher data rates, hardware limitations can prevent implementation of relatively complex satellite communication schemes with features such as coded modulation, shaping codes, multi-dimensional constellations, and non-equiprobable signaling. The present invention addresses simpler and more easily realized communication scheme using two-dimensional, uncoded modulation with equiprobable signals.
Previous work on two-dimensional constellation design has generally not associated the constellation with a particular hardware architecture. Such work has therefore tended to emphasize power measures related to the transmitter output (RF) power, such as the ratio at the receiver of the bit energy to the noise power density, Eb/N0. However, the transmitter prime power is a more useful criterion because it directly corresponds to the energy drain on the satellite""s batteries. Unless the constellation design is assumed fixed, the transmitter prime power and the output power are not linearly related because the efficiency with which the transmitter converts the former to the latter depends on the particular constellation employed.
Constellation design methods are discussed in articles by: R. W. Lucky et al. entitled xe2x80x9cOn the optimum performance of N-ary systems having two degrees of freedomxe2x80x9d, IRE Transactions on Communications Systems, 10:185-192. June 1962; S. Moskowitz, entitled xe2x80x9cSignal design for coherent M-ary communication systems using stochastic gradients. In Proceedings of the 5th Hawaii International Conference on System Science, pages 171-173, Honolulu. Hi., January 1972; G. J. Foschini, et al. entitled Optimization of two-dimensional signal constellations in the presence of Gaussian noisexe2x80x9d, IREE Trans. on Communications COM-22:28-38. January 1974; and S. Emami et al. entitled xe2x80x9cSignal selection for non-symmetric sources in two dimensionsxe2x80x9d, in Southeastcon, pages 461-464, Birmingham, Ala. April 1992.
The constellation design methods discussed in articles attempt to minimize the probability of symbol error while constraining the average or peak value of Eb/N0, usually by following an approximate gradient descent and renormalizing the symbol locations after each gradient update step in order to satisfy the power constraint. These methods are not applicable in the current context, because the hardware architecture restricts the symbols to a set of discrete locations that cannot be continuously adjusted, but only allowed or disallowed. In an article by J. Salz, et al. entitled xe2x80x9cData transmission by combined AM and PMxe2x80x9d, in The Bell System Technical Journal, 50(7):7399-2419, September 1971, the symbol space was assumed to be quantized to Na equally spaced amplitude levels and Np equally spaced phases per amplitude level. The corresponding constellations, dense near the origin and increasingly sparse away from it, radically differ from those discussed herein. Furthermore, the topic of pruning was not discussed in the Salz, et al. article, as K was always assumed to equal NaNp, where K is the number of constellation symbols.
Coding for certain non-rectangular constellations was examined in an article by T. Yamazato, et al. entitled xe2x80x9cAn arrangement technique of Gray-code table for signal constellation of modified QAM and triangular-shaped signal setxe2x80x9d, in IEICE Transactions, E 74(9): 2579-2585, September 1991. In that work, a one-to-one correspondence between the symbols of a rectangular, Gray-coded constellation and a non-rectangular constellation was established by xe2x80x9cmanuallyxe2x80x9d warping and transferring the symbols to lie on top of each other. Although this unsystematic technique may sometimes provide a good initial coding that can then be improved by the xe2x80x9cswappingxe2x80x9d coding algorithm developed herein, the technique seems ill-suited to irregularly shaped constellations.
In the above cited Foschini, et al. article, it was shown that under a constraint on the average value of Eb/N0, the hexagonal or xe2x80x9choneycombxe2x80x9d constellation is asymptotically optimal as the number of symbols goes to infinity. In a hexagonal constellation, the symbols are located at the centers of densely packed hexagons. For various finite constellation sizes, the superiority of the hexagonal constellation over other standard constellations (e.g., circular, cross, octagonal, pentagonal, rectangular, and triangular constellations) has also been demonstrated as discussed in articles by: K. Kawai, et al., entitled xe2x80x9cOptimum combination of amplitude and phase modulation scheme and its application to data transmission modemxe2x80x9d, in Proceedings of IEEE International conference on communications, pages 29.6-29.11, Phil., Pa., June 1972; M. K. Simon et al. entitled xe2x80x9cHexagonal multiple phase-and-amplitude-shift-keyed signal setsxe2x80x9d, in IEEE Trans. on Communications, COM-71:1108-1115, October 1973; and G. D. Forney, Jr., et al., entitled xe2x80x9cEfficient modulation for band-limited channels. IEEE Trans. on Selected Areas in Communications, 2(5):632-647, September 1984.
Accordingly, it is an objective of the present invention to provide for constellation pruning and partial Gray coding methods for use with a transmitter hardware architecture that implements multiple gigabit/second satellite communications.
To meet the above and other objectives, the present invention provides for improved constellation pruning and partial Gray coding methods. Such methods may be employed with a transmitter hardware architecture that implements multiple gigabit/second satellite communications. The transmitter hardware, which implements two-dimensional, uncoded signaling for equiprobable signals, comprises an array of interchangeable transmitter stages. Each stage of the transmitter includes a constant-signal phase shift keying (PSK) modulator and can generate multiple phase states. By way of example, stages with four or six states produce rectangular QAM or hexagonal constellations of symbols, respectively.
A more complex variant of the transmitter design also implements on-off keying, wherein any stage may be keyed into an additional xe2x80x9coffxe2x80x9d state, providing increased power efficiency. In the off state, the supply current to the power amplifier of a stage is interrupted and the power consumed by the stage is essentially zero. When several stages destructively interfere with each other, they are keyed into their off states to save power without altering the output of the transmitter.
Using this transmitter architecture, hexagonal constellations, for example, are produced when the transmitter stages have six phase states, and QAM (quadrature amplitude modulation or rectangular) constellations when there are four phase states.
Designing the transmitter requires determining the number of phase states per stage, the number of stages, whether to enable on-off keying, and which symbols to xe2x80x9cprunexe2x80x9d from excessively large constellations. Two design optimization criteria used in implementing the present invention are the transmitter prime power and the bit error rate. Although the transmitter output power is a more conventional criterion, the transmitter prime power is preferable because it directly relates to the amount of power consumed by the transmitter.
For binary communications, the pruned constellation is coded by assigning binary sequences to its symbols. Computationally efficient strategies for pruning and coding, provided by the present invention, yield coded hexagonal constellations, for example, that outperform traditional Gray-coded QAM constellations. In some cases, judicious selection of the design parameters can reduce the transmitter prime power by over 3 dB.
Signals generated using the transmitter correspond to symbols located in a two-dimensional space. The set of possible symbols, each of which is specified by its amplitude and phase or by its real and imaginary components, is termed the constellation. If it is desired that the transmitted symbol (signal) convey K bits of information, the constellation has size 2K, as the symbols can then be xe2x80x9ccodedxe2x80x9d or put into one-to-one correspondence with the set of binary sequences of length K. In general, the constellation size is larger than necessary, and some of the symbols must be disallowed or xe2x80x9cprunedxe2x80x9d from the constellation. Two performance measures are particularly important when optimizing the constellation design: the transmitter prime power (the power consumed by the transmitter), and the bit error rate when the received symbols are corrupted by Gaussian noise. Computationally efficient pruning and coding strategies based on these measures are provided by the present invention.
In particular, the present invention provides for a method of pruning and coding constellation transmitted by the transmitter. Pruning comprises the following steps. The approximate symbol error rate (SER) for each possible subconstellation is determined. Then, each of the subconstellations are scaled in terms of power so that each subconstellation has the same symbol error rate. Then, the subconstellation that requires the least amount of power to generate is chosen as the pruned constellation for use by the transmitter. After the pruned constellation is determined, it is then coded a single time, and this coded pruned constellation is used by the transmitter.
In coding the selected pruned constellation, (a) sequences to constellation symbols are randomly assigned. (b) A bit error rate for the constellation is then computed taking into account only nearest neighbor errors. (c) A predetermined number of constellation sequences are then switched. (d) A new bit error rate for the constellation is computed taking into account only nearest neighbor errors. (e) If the new bit error rate is reduced from the previous one, then the new mapping of sequences to symbols is retained; Otherwise the mapping prior to the switch is retained. (f) All possible switches are tested and whenever a switch is accepted, then testing begins again. All possible switches are tested until no switches are made. The predetermined number of constellation sequences that are simuntaneously switched may be two, three four, five, etc.
Alternatively, coding the selected pruned constellation may be accomplished by (a) selecting a point of an uncoded constellation. (b) The nearest neighbors of the selected point is found. (c) All possible sequences that can be used are evaluated, and an unused sequence from the available sequences that produces the best bit error rate is selected. (d) The remainder of the unused sequences is recorded. (e) The next nearest neighbors of the selected point are coded. (f) Steps (a) through (e) for each remaining point of the constellation are repeated.(g) If it is determined that the bit error rate is exceeded, indicating that a poor selection was made in step (c), the most recent decision is rejected and steps (c) through (e) are repeated until a code is selected that produces an acceptable bit error rate. (h) If it is determined in step (g) that none of the unused sequences produces an acceptable bit error rate, the next prior decision is rejected and steps (c) through (e) are repeated until a code is selected that produces an acceptable bit error rate. (i) Steps (a) through (h) are repeated for each remaining point of the constellation.